1. Numerical Weather Forecast Model (governing equations)
Momentum equations
Mass continuity equation
Moisture equation
Ideal gas law
Thermodynamic equation

2. Numerical Weather Forecast Model (governing equations)
Vertical momentum equation
Non-hydrostatic
Hydrostatic assumption (large scale phenomena):

3. Numerical Weather Forecast Model
Discretized in space (mesh)
Model grid points (boxes) x j
J-1
J+1
J-2 n
n-1
n+1
n-2 t
Discretized in time

4. Numerical Weather Forecast Model
Let’s simplify the equation to for now
How do we discretize it? j
J-1
J+1
J-2

5. Numerical Weather Forecast Model
B.C. u u
n=2
n=1
n=0 u
j=1
I.C.
j=N

6. Descretization
y=x2
Resolutions: 1
True solution

7. Numerical Weather Forecast Model
Global model vs. regional model
Regional
model

8. Weather Forecast WRF: Weather Research and Forecasting model
I.C.
B.C.
Outputs
Preprocessor
model
Terrain
Data
AVN,
ETA,…
WRF: Weather Research and Forecasting model

9. WRF Hurricane Isabel (2003) Usually forecast is not this good!
New KF, YSU, Purdue Lin, 10km
Radar observation
Model forecast
Usually forecast is not this good!
WRF

10. MM5 Model Configuration
Global Data: NCEP GDAS
Domain km km km
Physic schemes:
48-h simulation from 00Z 17 July 1997 .
• Betts-Miller convective scheme
• Blackadar PBL
• Mixed phase microphysics
• Simple radiation

11. Hurricane Danny Initial Conditions
(SLP, 950 mb wind vectors)
(950 mb moisture)
OBS mb .
Analysis mb
Reanalysis from global model

12. Forecasted Results  00Z/17/07- 00Z/19/07  Sea Level Pressure (SLP)
(hPa)
Time (hr)
obs
SLP at Storm Center
model
19.5mb (bad enough to scare you?)

13. Forecast Errors WRF: Weather Research and Forecasting model
I.C.
B.C.
Outputs
Preprocessor
model
Terrain
Data
AVN,
ETA,…
WRF: Weather Research and Forecasting model

14. Forecast Errors Model Errors: Dynamics (numerical schemes)
Physics parameterization
Resolution
2. Initial and boundary conditions (I.C/B.C.) error

15. Discretization (resolution)
Resolutions: 1
0.1
0.01 (more accurate)
Usually Dt is
proportional
to Dx.
True solution
Dx = 0.1
Dx = 0.01
=> the higher
the resolution,
the more
the computational time!
Dx =1

16. Problems of I.C. Reanalysis data – coarse resolution  Errors in I.C.
 Lack of mesoscale features in I.C.
 Model spin-up problem

17. WRF Model Flow Chart Data Assimilation Improved IC/BC Preprocessor
OBServations
Improved IC/BC
I.C.
B.C.
Outputs
Preprocessor
model
Terrain
Data
AVN,
ETA,…

18. Conventional Observations
12z
65 upper air soundings, 866 surface stations

19. Problems Reanalysis data – coarse resolution  Error in I.C
 Lack of mesoscale features in I.C.
 Model spin-up problem
Coarse resolution of World Meteorological Organization Upper-air radiosondes
 Twice a day
 Several hundred km resolution
Conventional data sparse areas
 Ocean
 Antarctic

20. Need Unconventional Observations
Remote Sensing Data
QuikSCAT

21. Using observations to improve model I.C. and B.C.
Objective Analysis
Using observations to improve model I.C. and B.C. j
J-1
J+1
Model grid points
Observations (obs) x
k-1 k
k+1 R r
J-2
radius of influence
Cressman method
: Weighting coefficient
R: radius of influence
r : distance between obs and model grid point j

22. Estimation and Data Assimilation
Suppose
Tm = 18 C (model temperature)
To = 21 C (observed temperature)
Suppose
m = 2 C (model error)
o = 1 C (observational error)
T is sought as: T = a Tm + b To
Such that the expected error: E{ ( T-Tt )2 } is minimal
Cost function

23. Optimal Estimation Introduction T = a Tm + b To Optimal solution o2
o2 + m2
m2
o2
a =
T = Tm ( To – Tm )
Optimal nudging
coefficient
Tm = 18 C (model)
To = 21 C (obs)
m = 2 C (model)
o = 1 C (obs)
T = 20.4 o C

24. Study: Danny, 1997 To comcompare two different
approaches for assimilating SSM/I data
 Retrieved products:
Total precipitable water (TPW)
Sea surface wind (SSW)
 Raw measurements:
Brightness temperature (Tb)

25. MM5 Model Configuration
Study: Danny
MM5 Model Configuration
Global Data: NCEP GDAS
Domain km km km
Physic:
48-h simulation from 00Z 17 July 1997 .
• Betts-Miller convective scheme
• Blackadar PBL
• Mixed phase microphysics
• Simple radiation

26. WRF Model Flow Chart Data Assimilation Improved IC/BC Preprocessor
OBServations
Improved IC/BC
I.C.
B.C.
Outputs
Preprocessor
model
Terrain
Data
AVN,
ETA,…

27. MM5 Data assimilation system
Study: Danny
SSM/I Data
Experiments
NCEP
GDAS
I.C.
CONTROL
Water vapor
Surface wind
I.C. RV
Irradiance
I.C. TB
MM5 Data assimilation system
1st guess

28. Study: Danny
Simulation Results
SLP at Storm Center
19.5mb

29. Study: Danny
Simulation Results
SLP at Storm Center
19.5mb

30. Simulation Rainfall First 12-h accumulated rainfall 0-1 hr 9~10 hr
Study: Danny
SSM/I Data
Simulation Rainfall
First 12-h accumulated rainfall
No SSM/I (CONTROL)
SSM/I (TB)
0-1 hr
9~10 hr